The short answer is yes! Boiling vegetables does remove nutrients, as well as pesticides from the product. However, you will still gain some nutritional value from eating them.
Cooking, boiling or baking can strip nutrients from foods, however, we must do it to keep some foods safe. It is primarily the hot water that strips nutrients from vegetables. Through the boiling process some of the nutrients are stripped away and lost, but much remains.
It is the hot water that causes the nutrients to leach out. This is also true of microwaving your food, but that is another topic.
Boiling tenderizes vegetables and prepares them in an appealing way, but fresh is the best if you want to obtain all of the nutrients that are offered. Vegetables lose a lot of their vitamins when boiled.
A good example is broccoli. Naturally, broccoli contains a large array of antioxidants and cancer-fighting properties. When boiled, the broccoli loses a lot of these nutrients. The best way to prepare broccoli, as with most vegetables, in a way that retains nutrients is to steam. Steaming tenderizes the veggies but keeps the nutrients intact.
While boiling vegetables does strip away any pesticides and toxins from the food, it is not always the best way to prepare a nutritious meal. Boiling vegetables does remove nutrients and it is important to keep this in mind when preparing.